#8550891

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4132 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,6,7)(3,11,12,8)(4,14,15,9)(5,10,16,13)(18,23)(19,25)(24,30)(26,31)(33,34,39,38)(35,40,42,37)(36,41,45,46)(43,47,48,44);(2,7)(3,12)(9,14)(10,13)(17,18,22,23)(19,21,25,29)(20,24,27,30)(26,28,31,32)(33,35,39,42)(34,40,38,37)(36,43,45,48)(41,47,46,44);(2,7)(4,15)(8,11)(10,13)(20,27)(24,30)(26,31)(28,32)(35,42)(37,40)(43,48)(44,47);(1,3,6,12)(2,8,7,11)(4,13,15,10)(5,14,16,9)(17,19)(18,21)(20,26)(22,25)(23,29)(24,28)(27,31)(30,32)(33,36)(34,41)(35,43)(37,44)(38,46)(39,45)(40,47)(42,48);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,11,16,8)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)(33,37)(34,35)(36,44)(38,42)(39,40)(41,43)(45,47)(46,48);(1,5)(2,10)(3,9)(4,11)(6,16)(7,13)(8,15)(12,14)(17,21)(18,25)(19,23)(20,28)(22,29)(24,31)(26,30)(27,32)(33,38,39,34)(35,37,42,40)(36,41,45,46)(43,47,48,44)

Polycyclic presentation:

2^9|a^2=hi,b^2=ghi,d^2=h,e^2=h,f^2=i,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8458087);